Non-Commutativity of Multiplication of Matrices

Matrix multiplication is widely used to find the solution in linear systems of equations. To multiply the two matrices, it is necessary that the two matrices should be compatible with each other for multiplication. The operations on matrices are the same as algebraic operations like addition, subtraction, and multiplication. Unlike addition and subtraction, it is not necessary for two matrices to have the same order for matrix multiplication. In this article, we will learn about the basics of matrix multiplication and the non-commutative property of matrices.

Matrix multiplication 

To perform matrix multiplication, we should always ensure that the two matrices follow the basic principle—the number of columns of the first matrix should be equal to the number of rows of the second matrix. For a given matrix A and B, the elements of new matrix C that is formed after multiplying the given two matrices are given by –

cij = ai1b1j + ai2b2j + ….+ ainbnj = k=1naikbkj

Steps to find matrix multiplication 

For finding the elements of a matrix obtained by multiplying the given two matrices. The steps are as follows:

• Elements in row 1 and column 1 of matrix A will be multiplied with elements in row 1 and column 1 of matrix B (a11 × b11).
• Element in row 1 and column 2 of matrix A will be multiplied with elements in row 2 and column 1 of matrix B (a12 × b21).
• Element in row 1 and column 3 of matrix A will be multiplied with elements in row 3 and column 1 of matrix B (a13 × b31).
• The addition of both products calculated above will give the elements of row 1 and column 1 of matrix C (c11 = a11 × b11 + a12 × b21 + a13 x b31).

Order of the matrix obtained by multiplying the two matrices 

The order of the matrix obtained by multiplying the two compatible matrices will be equal to the number of the rows of matrix 1 and the number of columns of matrix 2.

For example,

A = 3 6 9 4 2×2 B = 1 0 5 4 2 14 2 x 3 

There, the order of the matrix A is 2×2 and the order of matrix B is 2×3. Thus, the number of rows of matrix A is 2 and the number of columns of matrix B is 3. Thus, the new matrix formed would be the order of 2×3.

Non-commutativity of matrix multiplication

The commutative property over multiplication is a × b = b × a Thus, this property tells us that the order of the numbers does not change the output of the function. In matrix multiplication, we have seen that whenever we multiply two matrices, we may or may not have the same order. We get the new matrix whose order is equal to the number of the rows of matrix 1 and the number of columns of matrix 2. 

The matrix multiplication is generally not commutative because when we multiply the two matrices, the elements of first row of matrix I is being multiplied by the elements of first column of matrix II due to which changing the order will change the corresponding elements of matrix I and matrix II. However, for some matrix multiplication, for example, for null matrices or for identity matrices this property holds but is not applicable on all the matrices.

Proof of non-commutativity of matrices

Let two matrices A and B such that

A = 6 3 10 1 0 5 2 x 3  B = 0 2 1 3 4 6 3 x 2

AB = 6 3 10 1 0 5   .  0 2 1 3 4 6

AB = 0+3+40 12+9+60 0+0+20 2+0+30

AB = 43 81 20 32

Now,

BA = 0 2 1 3 4 6 .  6 3 10 1 0 5                 

BA = 2 0 10 9 3 25 30 12 70       

Here, we can see that AB ≠BA, which shows the non-commutativity of matrix multiplication.

Conclusion

Matrix multiplication is not commutative over multiplication, as commutativity is defined as a x b = b x a. But in the case of matrix multiplication, whenever we multiply two matrices, the new matrix may have some different order depending upon the number of rows and columns of the first and second matrix, respectively. However, in some cases, the matrix multiplication is commutative. Those are some exceptional cases.